### Using Progressions to explore if there are infinite Pythagorean Triples Containing Two Primes

Iain Preston
2020 Figshare
By applying the Theorem regarding Primitive solutions of the Pythagorean equation, this paper initially sets out how a group of Pythagorean triples, that is triples (x, y, z) each comprising of positive integers such that x2 + y2 = z2, are constructed from consecutive positive integers, n and n + 1. This paper then explains how progressions can be formulated for the sequences of the odd numbers in y and z (having categorised x, y and z using the Convention for Pythagorean triples) which arise
more » ... om those triples. From the results it was apparent that in this group of triples, the values of y can be represented by the arithmetic progression 1 + 2n. As such, the progression for the values of y contains the entire set of odd numbers from 3 onwards and hence by definition also contains the entire set of prime numbers, except for 2. Similarly it was apparent that the values of z take the form "4k + 1", but each term in the progression of the z values was found to be based on the previous term, and therefore the progression could be represented by the formula zn = zn-1 + 4n. Finally, following a review of the group of triples for the occurrences of primes in either or both of y and z and for values of z < 1 million, 42 triples were identified which contained pairs of primes. Based on all the findings, an explanation is put forward to support why there can be infinite primitive Pythagorean triples which contain pairs of primes. It is, of course, unlikely any new ideas or mathematics have been developed in this paper, so it will not meet the criteria for journal publication. It is therefore for interest only and should not be relied on in any way.